Sequence independent lifting for mixed knapsack problems with GUB constraints

Abstract

In this paper, we consider the semi-continuous knapsack problem with generalized upper bound constraints on binary variables. We prove that generalized flow cover inequalities are valid in this setting and, under mild assumptions, are facet-defining inequalities for the entire problem. We then focus on simultaneous lifting of pairs of variables. The associated lifting problem naturally induces multidimensional lifting functions, and we prove that a simple relaxation in a restricted domain is a superadditive function. Furthermore, we also prove that this approximation is, under extra requirements, the optimal lifting function. We then analyze the separation problem in two phases. First, finding a seed inequality, and second, select the inequality to be added. In the first step we evaluate both exact and heuristic methods. The second step is necessary because the proposed lifting procedure is simultaneous; from where our class of lifted inequalities might contain an exponential number of these. We choose a strategy of maximizing the resulting violation. Finally, we test this class of inequalities using instances arising from electrical planning problems. Our tests show that the proposed class of inequalities is strong in the sense that the addition of these inequalities closes, on average, 57.70 % of the root integrality gap and 97.70 % of the relative gap while adding less than three cuts on average.